Twisting with a twist: supramolecular helix fluctuations in chiral
nematics

Abstract

Most theoretical descriptions of lyotropic cholesteric liquid crystals to date focus on homogeneous systems in which the rod concentration, as opposed to the rod orientation, is uniform. In this work, we build upon the Onsager-Straley theory for twisted nematics and study the effect of weak concentration gradients, generated by some external potential, on the cholesteric twist. We apply our theory to chiral nematics of nanohelices in which the supramolecular helix sense is known to spontaneously change sign upon variation of particle concentration, passing through a so-called compensation point at which the mesoscopic twist vanishes. We show that the imposed field offers exquisite control of the handedness and magnitude of the helicoidal director field, even at weak field strengths. Within the same framework we also quantify the director fluctuation spectrum and find evidence for a correlation length diverging at the compensation point.

Chiral intermolecular forces are essential for stabilizing the building blocks of life (e.g. the amino acids that make up DNA) and play an important role in key biological processes. Condensed phases composed of chiral constituents exhibit a much richer phase morphology than their non-chiral counterparts. Examples are liquid crystal mesophases consisting of elongated chiral mesogens which may form twisted nematic Borshch et al. (2013) or cubic blue phases Coles and Pivnenko (2005) whose chirality-induced periodic mesostructure endows them with special opto-electronic properties de Gennes and Prost (1993). These materials find important applications in electronic displays, smart windows, optical switches, photonics and cosmetic products.

Recent theoretical and simulation studies utilizing coarse-grained models for curled hard cylinders Kolli et al. (2014); Dussi et al. (2015); Wensink and Morales-Anda (2015) or helical patchy rods Emelyanenko et al. (2000); Wensink (2014); Ruzicka and Wensink (2016); Kuhnhold and Schilling (2016) have shed new light on how molecular chirality translates into various macroscopic structures. Most of the focus has been on cholesteric liquid crystals. These structures are essentially nematic (no long-range positional order) but the local director exhibits a helical precession, characterized by an intrinsic length scale, the helical pitch P, and handedness (left-handed, LH or right-handed RH, see Fig. 1).
One of the remarkable findings emerging from these studies is that the cholesteric sense (a left- or right-handed twist) is not only dictated by the chirality at the particle scale Katsonis et al. (2012) but also by the thermodynamic state of the system Dussi et al. (2015). Helical mesogens with a certain prescribed molecular helicity may undergo spontaneous sense inversions by subtle variations of the overall particle concentration, pressure or temperature Wensink (2014). Temperature-induced sense inversions are not uncommon in certain thermotropic systems Slaney et al. (1992); Yamagishi et al. (1990); Toriumi et al. (1983); Watanabe and Nagase (1988); Stegemeyer et al. (1989),
but their origin is unclear. Most likely, subtle modifications in the molecular chirality or solvent conditions upon variation of temperature are at the core of these trends. The supramolecular handedness may also be controlled using photosensitive chiral dopants Mathews et al. (2010). Furthermore, mixing components each with a different sign and magnitude of the molecular chirality may lead to situations where the global twist vanishes. These particular states are usually referred to as compensated or racemic cholesterics Muller and Stegemeyer (1973) and re-emerge in nanohelix cholesterics at the inversion point where the handedness changes sign and the supramolecular twist becomes zero Lubensky et al. (1996). The lyotropic case is surprising in that spontaneous sense inversions happen at fixed internal chirality and interaction range. The inversions are brought about solely by a subtle interplay between concentration and (local) particle alignment Dussi et al. (2015); Wensink (2014). A further experimental exploration of these sense inversions, which have to date not been identified in the biofibril suspensions listed above, is desirable as it may open up new possibilities to tune the optical properties of lyotropic cholesteric materials Mitov (2012).

Figure 1: (a) Simulation snapshot of a left-handed (LH) cholesteric phase of helical patchy cylinders. The vertical system dimension corresponds to half the pitch P of the helical director field (rod orientations are color coded). Reprinted from Ref. Ruzicka and Wensink (2016). An external field Uext coupling to the local particle concentration creates a non-uniform density along the pitch direction z. (b) The particles consist of a soft helical potential (indicated by the green dots) with pitch p wrapped around the surface of a cylindrical hard core (in red). (c) The mean-field chiral potential between a rod pair at fixed centre-of-mass distance depends on the interrod angle γ and may display a single-mimimum (ℓ=1) or double-minimum behavior (ℓ=3), depending on the sign and amplitude of the molecular pitch p. Helices with ℓ=1 are LH, those with ℓ=3 possess a RH symmetry Wensink (2014).

In this work we take a closer look at fluctuations in the supramolecular twist in case the particle concentration is no longer spatially uniform but subject to a weak modulation induced by some external field. The main questions we set out to address are the following. First: How does a weak modulation of the particle concentration couple to the local director twist and can we exploit this to generate more complex non-uniform periodic twist profiles? The second question relates to fluctuations in the supramolecular twist induced by thermal motion; Is there a diverging length scale associated with director fluctuations upon approach of the compensation point, and, if so, how does this correlation length depend on particle concentration?

To address both issues we revisit Onsager’s second-virial theory Onsager (1949) for nematic phases of slender hard rods, supplemented with Straley’s extension Straley (1976a) to account for the effect of a non-uniform (e.g. twisted) director field. We further generalize the framework toward systems with a weak gradient in the particle concentration. The ensuing theory is essentially a hybrid square-gradient theory accounting for the subtle coupling between concentration and director deformations mediated by the local particle orientations. An important advantage of using the Onsager-Straley approach, in contrast to some of the more expansive density functional theories formulated for chiral nematics Belli et al. (2014); Allen (2016); Tortora and Doye (2017), is the direct connection with the pair potential of the (helical) nanorods. No experimental or simulation input is required to quantify the elastic properties of the system since they are intrinsically calculable within the theory. The approach thus enables us to predict the fluctuation spectrum of nanohelix cholesterics on a microscopic footing.

The manuscript is organized as follows. We begin in Section II by laying out a simple square-gradient formalism derived from the Onsager-Straley theory for chiral nematics. The required microscopic input parameters are discussed in Section III based on a rigid hard rod model supplemented with some tractable helical potential mimicking the twist propensity of a pair of soft helical filaments. The implications of a weak concentration modulation along the pitch axis on the local director twist is investigated in Section IV and the director fluctuation spectra will be analyzed and discussed in detail in Section V. The main conclusions drawn from this study will be formulated in the final Section.

The starting point of our analysis is Onsager’s classical second-virial theory Onsager (1949) designed for fluid phases of infinitely slender, rigid filaments where interactions involving more than two particles are highly improbable. The excess free energy Fex in units of the thermal energy kBT (with temperature T and Boltzmann’s constant kB) may be generalized for inhomogeneous systems and formally reads Poniewierski and Holyst (1988); Allen et al. (1993)

Fex[ρ]kBT=−12∬dr1dr2⟨⟨ρ(r1,^ω1⋅^n(r1))ρ(r2,^ω2⋅^n(r2))Φ⟩⟩,

(1)

where the Mayer function Φ=e−U/kBT−1 relates to the pair potential U between two particles (“1” and “2”). It depends explicitly on their mutual orientation, indicated by the unit vectors ^ωi, and their centre-of-mass distance r1−r2. The one-body density ρ expresses the probability to find a rod with centre-of-mass at position r and orientation ^ω with respect to a spatially varying director field ^n(r). Brackets denote a double angular average ⟨⟨⋅⟩⟩=∫d^ω1∫d^ω2. Our working assumption is that gradients in the nematic director as well as in the particle concentration extend over distances far greater than the typical particle scale. Defining new coordinates R=(r1+r2)/2 and Δr=r1−r2 we may expand ρ up to linear order in Δr. This yields two gradient contributions, one for the concentation and a second one describing spatial variations of the director field Straley (1976a), respectively

ρ(ri,^n(ri)⋅^ωi)

=ρ(R,^ωi⋅^n(R))

±(Δr2⋅∇R)ρ(R,^ωi⋅^n(R))

(2)

for i=1(+),2(−), in terms of the partial derivative of the one-body density with respect to orientation ˙ρ(R,^ω⋅^n(R))=∂ρ(R,^ω⋅^n(R))/∂(^ω⋅^n(R)). In the following we shall focus on a weakly twisted director field with a helical axis fixed along the z−direction of the laboratory frame which we denote by Cartesian coordinates (X,Y,Z). The twist deformation then reads ^n(Z)≈(1,φ(Z),0) with a non-uniform twist angle φ(Z) (|∇φ|≪1). An expansion of the free energy per unit surface A up to second order in the gradients gives

FexAkBT=∫dZ⟨⟨{12M0ρ(Z,^ω1)ρ(Z,^ω2)+12M1ρ(Z,^ω1)ω2y˙ρ(Z,^ω2)∇φ(Z)

(3)

In deriving the above, we have imposed mirror reflection symmetry, ρ(Z,^ω)=ρ(−Z,^ω) by requiring that all linear terms ∇ρ be zero.
The kernels Mn refer to the n-th moment of the Mayer function and are defined as

Mn(^ω1,^ω2)=−∫dΔr(Δr⋅^z)nΦ(Δr,^ω1,^ω2).

(4)

These quantities depend explicitly on the mutual particle orientation of a rod pair and provide the key microscopic input of our theory. The kernels will be specified in the next Section.
The odd term M1 is only non-zero if the rod interactions are chiral in which case the direction of twist deformation matters, i.e., ∇φ≠−∇φ. For achiral particles all terms linear in ∇φ vanish. The excess term involves pair-interactions only and is merely approximate at elevated particle densities. The remaining free energy contributions on the other hand are exact and represent the free energy of an ideal gas of rodlike particles via

Fid[ρ]AkBT=∫dZ⟨ρ(Z,^ω)[lnVρ(Z,^ω)−1+Uext(Z,^ω)]⟩,

(5)

where the last term imparts the effect of some externally imposed potential Uext and V is an immaterial thermal volume containing contributions from the rotational momenta of the particles.
The next step is to minimize the total free energy with respect to the density ρ(Z,^ω) while assuming the density to be unaffected by the weak director twist. This is done by means of a functional minimization δδρ(Z,^ω)[F−μ∫dZ⟨ρ(Z,^ω)⟩]∇φ=0=0, in terms of a chemical potential μ acting as a Lagrange multiplier to ensure a fixed particle number; ∫dZ⟨ρ(Z,^ω)⟩=N/A. The result is an Euler-Lagrange equation for the one-body density which can be recast as a Boltzmann exponent

ρ(Z,^ω1)

=1Vexp(−β[US(Z,^ω1)−μ]),

(6)

in terms of a self-consistent field US combining some effective internal potential due to rod-rod correlations and the external one

US(Z,^ω1)=⟨M0ρ(Z,^ω2)+14M2∇2ρ(Z,^ω2)⟩^ω2+Uext(Z,^ω1).

(7)

Minimization of the total free energy with respect to the twist deformation δF/δ∇φ(Z)=0 yields for the equilibrium twist

∇φ(Z)=Kt(Z)K2(Z)

(8)

where the coefficients relate to a weighted double angular average of the kernels

βKt(Z)

=−12⟨⟨M1w2yρ(Z,^ω1)˙ρ(Z,^ω2)⟩⟩,

βK2(Z)

=12⟨⟨w1yw2yM2˙ρ(Z,^ω1)˙ρ(Z,^ω2)⟩⟩.

(9)

The results for systems with a uniform particle concentration ρ0 are easily retrieved by setting ρ(Z,^ω)=ρ0f0(^ω). The local orientation distribution function (ODF) f0 then follows from f(^ω1)=Nexp(−ρ0⟨M0(^ω1,^ω2)f0(^ω2)⟩^ω2) with the constant N ensuring normalization via ⟨f0(^ω)⟩=1. Likewise, the two coefficients Eq. (9) reduce to the familiar torque-field and the (Frank) twist elastic constants, defined as Allen et al. (1993)

βKt

=−ρ202⟨⟨M1w2yf0(^ω1)˙f0(^ω2)⟩⟩,

βK2

=ρ202⟨⟨w1yw2yM2˙f0(^ω1)˙f0(^ω2)⟩⟩.

(10)

The ratio of these two give a uniform twist deformation ∇φ(Z)=Kt/K2=q0 with q0 a wavenumber inversely proportional to the pitch of the cholesteric system. The more general expressions Eq. (6) and Eq. (9) enable us to compute the non-uniform twist profile of a cholesteric phase exposed to an external potential acting along the pitch direction. In Section IV, we shall take a closer look at the implications of a weak concentration gradients imposed by some arbirtrary external field (related to e.g. particle sedimentation, solvent evaporation, or the presence of a substrate). But first, we need to specify the microscopic interactions that underpin the stability of cholesteric order in suspensions of helical filaments.

Let us consider the interactions between a pair of hard cylindrical rods with length L and diameter D, each padded with some helical surface pattern, resembling a helical ‘patchy’ particle Ruzicka and Wensink (2016); Kuhnhold and Schilling (2016). For reasons of symmetry, the even kernels M0 and M2 featuring in the square-gradient free energy Eq. (3) only depend on the achiral hard cores. The Mayer function Φ yields -1 when the cores overlap and zero otherwise. For hard cylinders with infinite length-to-width ratio L/D→∞ the kernels correspond to the following (generalized) excluded volumes Odijk (1986); Wensink and Jackson (2009)

M0

∼2L2D|sinγ|,

M2

∼16L4D|sinγ|[(^ω1⋅^z)2+(^ω2⋅^z)2],

(11)

with |sinγ|=|^ω1×^ω2|. The odd kernel M1 depends on the specific chiral interaction Uc between the helical filaments and is strictly zero in the absence of chirality. For weakly chiral interactions (Uc≪kBT) it is justified to approximate Φ≈−βUc. To mimic the effective potential between soft helical filaments Wensink (2014) we propose the following simplified form

Uc∼εcg(Δr)(^ω1×^ω2⋅Δr){π2γccos(π2ℓγγc)|γ|≤γc0|γ|>γc.

(12)

This potential is intrinsically chiral since it is not invariant with respect to the inversion operation Δr→−Δr. The decay with increasing centre-of-mass distance is given by g(Δr). The pseudoscalar form (^ω1×^ω2⋅Δr) originally emerged from electric multipole expansions Goossens (1971); van der Meer et al. (1976)
but has since then been consistently used in simulation models to capture chiral interactions (whether caused by quantum-mechanical or steric factors) between non-spherical mesogens Varga and Jackson (2003); Memmer et al. (1993); Berardi et al. (1998); Germano et al. (2002); Wilson (2005). As for the remaining parameters, εc is an amplitude parameter and γc a cut-off value for the angle, such that Uc(γc)=0. Most importantly, ℓ=1,3,5… is an odd integer determining the number of local minima in Uc(γ). This is illustrated in Fig. 1(c). The case ℓ=1 produces a single minimum function imparting a uniform helix sense, whereas the double-minimum form for ℓ=3 gives rise to pitch inversion scenario where the cholesteric helix sense switches handedness upon changing the overall particle concentration of the cholesteric system. The kernel M1 can be approximated by introducing a cylindrical laboratory frame (Δr⊥,Δz)

M1

=−∫dΔr(Δr⋅^z)βUc(Δr,^ω1,^ω2)

∼−¯εcL4(^ω1×^ω2⋅Δ^z){π2γccos(π2ℓγγc)|γ|≤γc0|γ|>γc,

(13)

where the spatial integral over the decay function is now subsumed into some effective dimensionless chiral amplitude via

¯εc=πεckBTL−4∫∞0dΔr2⊥∫∞−∞dΔz(Δz)2g(Δr⊥,Δz).

(14)

The precise form of g(Δr) is not crucially important as long as convergence of the spatial integral is guaranteed and the condition ¯εc≪1 is met.
We emphasize that the definition of ¯εc makes the theory applicable to a wide range of cholesteric materials of rigid helical filaments where chiral forces are transmitted primarily by long-ranged, soft interactions rather than by steric forces related to particle shape Kolli et al. (2016).
For the case ℓ=3 the critical concentration at which a helical sense inversion occurs is inversely proportional to γc. In our calculations, we choose γc=0.5 in which case a pitch sense inversion occurs at a concentration of c0=ρ0L2D=17.84. The isotropic-cholesteric phase coexistence densities are located at c(I)0=4.189 and c(N)0=5.336Vroege and Lekkerkerker (1992). Some relevant numerical results for the pitch versus concentration have been compiled in Fig. 2. For the homogeneous systems, standard iteration routines utilizing an equidistant grid of relevant angles to discretize orientational space ^ω were employed to solve equations such as Eq. (6) Herzfeld et al. (1984); van Roij (2005).
In Fig. 2 two distinct scenarios are highlighted: a conventional one (ℓ=1) in which the pitch decreases monotonically with concentration, as routinely encountered in a wide range of bio-inspired cholesteric liquid crystals Dogic and
Fraden (2000b); DuPré and Duke (1975); Schütz et al. (2015); Belamie et al. (2004); Giraud-Guille et al. (2008). The second case (ℓ=3) relates to a pitch-inversion scenario where the twist suddenly changes handedness at a critical concentration and, associated with this, a critical degree of local nematic alignment Wensink (2014); Dussi et al. (2015); Ruzicka and Wensink (2016); Kuhnhold and Schilling (2016). The microscopic underpinning for this phenomenon resides in the double-minimum form of the chiral potential (see Fig. 1(c)). Since the two minima are located at opposite signs of the twist angle the global twist sense imparted by the chiral potential depends critically on the degree of nematic alignment ∼⟨⟨γ⟩⟩ along the director field, which is steered by particle concentration Wensink (2014).

Figure 2: (a) Local nematic order parameter S versus concentration for a lyotropic cholesteric of chiral rods. (b) Corresponding helical pitch (in units L/¯εc) and handedness for a system with a monotonically decreasing pitch (ℓ=1) and a system exhibiting a spontaneous inversion of the cholesteric handedness (ℓ=3). At the compensation point (c0≈18, S≈0.98) the supramolecular twist vanishes as indicated by a divergence of the pitch (vertical dotted line).

A rough estimate for ¯εc can be produced by assuming helical rods with some chiral charge pattern Rossi et al. (2011); Tombolato et al. (2006) with an effective total charge Qeff residing on the particle surface, so that chiral forces are mediated via some screened Coulomb potential Q2effλBexp(−κr)/r with λB the Bjerrum length and κ the Debye screening constant related chiefly to the ionic strength of the solvent. Using this in Eq. (14)
we write ¯εc as a simple product of Qeff and a number of (dimensionless) size ratios

¯εc∼Q2eff(λB/D)(D/L)3(κD)−2.

(15)

We may test the usefulness of this prediction by plugging in typical numbers for e.g. filamentous virus rods Tombolato et al. (2006). Taking order-of-magnitude estimates for the relevant size ratios, λB/D∼O(10−1), virus aspect ratio D/L∼O(10−2), effective surface charge Qeff∼O(103), and electrostatic screening κD∼O(1), yields ¯εc∼O(10−1). Similarly, reasonable estimates for cellulose nanocrystals (CNCs) Schütz et al. (2015) are: λB/D∼O(10−1), D/L∼O(10−2), Qeff∼O(102), and κD∼O(1) gives ¯εc∼O(10−1−10−2). Reading off typical values in Fig. 2b we obtain for the pitch length P∼(2π/q0)(L/¯εc)∼O(L/¯εc) so that P/L∼O(101−102). Given that nanorod contour lengths lie in the range L∼0.1−1 microns, the corresponding pitches amount to tens of microns, in full accordance with what is routinely measured in experiment.

Let us assume a small perturbation from the uniform particle concentration

ρ(Z,^ω)=ρ0f0(^ω)+δ^ρq(^ω)eiqZ,

(16)

imparted by some weak external periodic potential of the form Uext(Z)=^ueiqZ with amplitude ^u≪1 acting on the positional coordinates alone. Examples could be concentration gradients imposed by e.g. an laser-optical trap, a temperature gradient, solvent evaporation or particle sedimentation or induced by the presence of a substrate or interface. Linearising the Euler-Lagrange equation Eq. (6) we obtain a self-consistency equation for δ^ρq

−δ^ρq(^ω1)=ρ0f0(^ω1)[β^u+⟨(M0+q24M2)δ^ρq(^ω2)⟩^ω2],

(17)

for every mode q≠0. Inserting the perturbed one-body density Eq. (16) into the coefficients Eq. (9) and retaining contributions up to linear order allows us to write Kn(Z)∼Kn+δKneiqZ (n=t,2).The linear perturbations depend implicitly on particle concentration ρ0 and wavenumber q of the imposed concentration fluctuation (through Eq. (17)) and the orientational distributions via

δKt

=−ρ02[⟨⟨M1w2yf0(^ω1)δ˙^ρq(^ω2)⟩⟩

+⟨⟨M1w2yδ^ρq(^ω1)˙f0(^ω2)⟩⟩],

δK2

=ρ0⟨⟨M2w1yw2yδ˙^ρq(^ω1)˙f0(^ω2)⟩⟩.

(18)

The non-uniform twist then becomes up to linear order in δ^ρq

∇φ(Z)=q0+χeiqZ+O(δ^ρ2),

(19)

where q0=Kt/K2 is the helical wave-number of the uniform cholesteric phase.
The susceptibility χ=∂q0/∂|δ^ρq| has units of inverse length and expresses the non-trivial linear response of the pitch of a cholesteric nematic upon imposing a weak concentration fluctuation along the pitch direction. It reads

χ=δKt−q0δK2K2,

(20)

and is nonzero because the local rod orientations areaffected by the imposed density gradient. Solving Eq. (17) numerically we find a monotonic increase of χ with the field amplitude ^u and a negligible dependency on q in the weak-gradient regime q≪1.

Figure 3: Applying a weak external field of strength ^u (in units kBT) induces a concentration modulation along the pitch axis which distorts the uniform twist of the director field. The amplitude of the local twist deformation χ (which has units inverse length, ¯εc/L) is plotted as a function of the overall particle concentration c0. For the case ℓ=3 there is a point of zero response around c0≈11.1 (blue dot). The compensation point where the global twist vanishes (q0↓0) is indicated by blue vertical dotted line.

Eq. (19) tells us that the external field renders the local twist non-uniform and causes the nematic director field to adopt a more complicated helicoidal topology.
The director component perpendicular to the reference direction (x-axis) twists in the following way

^ny(Z)≈q0Z+χq−1sin(qZ),

(21)

In practice, in view of the square-gradient approximation underpinning Eq. (17) the wavelength of the imposed concentration wave should be small (q≪1) so that

^ny(Z)≈(q0+χ)Z,

(22)

independent of q. The evolution of the response χ as a function of the overall particle concentration is shown in Fig. 3. The response is simply monotonically increasing with c0 for the ℓ=1 scenario (without pitch inversion), while the case ℓ=3 exhibits a marked point of zero response at a density preceding the compensation point. At the zero point the effect of the applied field on the local twist vanishes. It roughly corresponds to the concentration where the derivative of the pitch with concentration becomes zero, ∂q0/∂c0→0 (blue dot in Fig. 2b). We stress, however, that the concentration-orientation coupling renders the response strongly non-linear so that χ does not obey a simple prescription χ∼∂q0∂c0δc0(u), with δc0(u) the field-induced change of the local concentration, one could have naively proposed.

At the compensation point, where the intrinsic twist vanishes (q0↓0), a global twist can be imposed by the external field.
Variation of the amplitude and sign of the external potential via u thus allows for a judicious tuning of the handedness and the pitch length of the helicoidal director field. This is illustrated in the bottom panel of Fig. 3. Typically, an imposed field strength of 0.01kBT suffices to bring about a change in the helical pitch of order χ−1∼O(L/¯εc) where ¯εc depends on the molecular details of the filaments responsible for transmitting chirality (see Eq. (14)).
Recalling the estimate ¯εc∼O(10−1−10−2) for typical chiral nanorods (Section III) we conclude that the impact of a weak concentration gradient on the pitch is expected to be quite significant.

In this Section we attempt to quantify the range and strength of thermal fluctuations the helicoidal director field experiences. We shall focus in particular on the behaviour of these fluctuations in the vicinity of the compensation point where the cholesteric twist vanishes. In contrast to most phenomenological theories put forward to date de Gennes and Prost (1993); Rey (2010); Yamashita (2004); Yoshioka et al. (2012), the Onsager-Straley theory enables us to gauge the elastic properties of the cholesteric from a microscopic standpoint and establish an explicit dependence of the fluctuation spectrum with respect to particle concentration. Let us consider the following perturbations of the helical director field

^nx(R)

=cos(q0Z+∑k⊥δ^qk⊥eik⊥⋅R)

^ny(R)

=sin(q0Z+∑k⊥δ^qk⊥eik⊥⋅R)

^nz(R)

=∑k∥δ^qk∥eik∥⋅R,

(23)

where the amplitude |δ^qk⊥|≪1 refers to a weak modulation of the linear twist φ(Z)=q0Z and |δ^qk∥|≪1 to a spatial perturbation of the pitch direction (along the z-axis). The change in excess free energy produced by a weak non-uniformity of the director field takes the following form Straley (1976b); Wensink and Jackson (2009)

FtwistkBT∼ρ22∫dR⟨⟨∫dΔr∂R(^ω2)Φf0(^ω1)˙f0(^ω2)⟩⟩

−ρ24∫dR⟨⟨∫dΔr∂R(^ω1)∂R(^ω2)Φ˙f0(^ω1)˙f0(^ω2)⟩⟩+⋯,

(24)

where ∂R(^ωi)=(Δr⋅∇R)^n(R)⋅^ωi. Ignoring the fluctuation terms, we easily retrieve the mean-field free energy of a weakly twisted cholesteric by inserting Eq. (23) and expanding up to quadratic order in q0 so that Ftwist/kBT=−Ktq0+12K2q20 (cf. Eq. (10)).
It is now fairly straightforward to work out the free energy change imparted by a weak spatial modulation of the helicoidal director field by inserting ^n(R) and retaining the leading order contributions for small amplitudes δ^qk⊥ and δ^qk∥. Focussing on the latter first, we obtain for the free energy change associated with longitudinal director fluctuations along the pitch direction

δF∥V∼12∑k∥{δ^q2k∥[k2∥,xK3+k2∥,yK2+k2∥,zK1]},

(25)

in terms of the splay (K1) and bend (K3) elastic constants, specified in the Appendix.
From the quadratic contribution we can infer the following fluctuation spectrum upon invoking the equipartition theorem de Gennes and Prost (1993)

⟨δ^q2k∥⟩∼kBTV(k2∥,xK3+k2∥,yK2+k2∥,zK1).

(26)

It suggests that fluctuations in the pitch direction decay algebraically, irrespective of the cholesteric twist q0.
A similar analysis produces the following spectrum for the transverse fluctuations (i.e. perpendicular to the pitch axis z) of the local nematic director

⟨δ^q2k⊥⟩∼kBTV(k2⊥,xK3+k2⊥,yK1+k2⊥,zK2+q20K∗),

(27)

where K∗>0 is an additional elastic constant specified in the Appendix. Taking the inverse Fourier transform (FT) of this expression we find that the transverse director fluctuations along the helicoidal axis decay exponentially

⟨δq⊥(Z)2⟩∼kBTV(K∗K2)12e−|Z|/ξzq0,

(28)

in terms of a correlation length

ξz∼(K2K∗q20)12∼(710π)121|q0|c0.

(29)

A similar behavior is found for the decay of transverse fluctuations measured along the local director (which is fixed along the x-axis of the lab frame). The approximation k2⊥,xK3+k2⊥,yK1≈k2⊥,xK3 seems justifiable for concentrated hard rod systems where the splay modulus is much smaller than the bend one (K1≪K3). Performing an inverse FT of Eq. (27) we obtain a similar exponential form ⟨δq⊥(X)2⟩∼kBTV−1(K∗K3)−12e−|X|/ξx/q0 whose amplitude now involves the bend modulus K3. The correlation length for transverse director fluctuations probed along the local director also diverges at the compensation point, albeit with a different concentration scaling than ξz

ξx∼(K3K∗q20)12∼15121|q0|.

(30)

Figure 4: Correlation length (in units L/¯εc) measuring the decay of director fluctuations transverse to the pitch axis probed along the pitch axis (ξz) and along the local nematic director (ξx). For the case ℓ=3 both length scales diverge at the compensation point where the global twist vanishes.

This correlation length is of the order of the helical pitch 1/q0 whereas ξz<ξx throughout the probed concentration range. The concentration dependence of these correlation lengths can be established in explicit form from the asymptotic results for the elastic constants of infinitely slender hard rods which have been compiled in the Appendix.
The expressions above clearly demonstrates that both correlation lengths and their respective amplitudes diverge at the compensation point (see Fig. 4). This suggests that the crossover from one handedness to the other upon changing the thermodynamic state (particle concentration or temperature) as reported in a number of recent studies Wensink (2014); Belli et al. (2014); Dussi et al. (2015); Ruzicka and Wensink (2016); Kuhnhold and Schilling (2016) constitutes some higher-order phase transition where director fluctuations diverge critically at the compensation point.

We have investigated in which way the supramolecular twist in a lyotropic cholesteric structure is affected by weak gradients in particle concentration as well as by thermal fluctuations. Our focus is on lyotropic assemblies of helical nanohelices where chiral torques are transmitted through some weak helical surface potential for which we propose a simple coarse-grained potential. This serves as the microscopic basis of an Onsager-Straley theory for twisted nematics which we have generalized to account for weak concentration gradients.
Applying a generic external potential acting only on the centre-of-mass coordinates induces a weak modulation of the concentration along the pitch direction. We show that the concentration gradients couple non-linearly to the cholesteric twist via the average rod orientations and demonstrate that spatially non-uniform twist patterns can be generated in this manner.
In case the system is near a so-called compensation point where the global twist but not the molecular chirality vanishes, a significant change in the pitch can be realized for weak potential amplitudes. This effect can be exploited to tune the supramolecular twist of lyotropic materials without the need to modify the molecular chirality, for instance, by changing the solvent conditions or temperature.

In the second part of this work we use the Onsager-Straley framework to identify how the twisted director field is affected by thermal fluctuations. Upon deriving the director fluctuation spectrum for nanohelix cholesterics we put forward an analytical expression relating the correlation length which measures the decay of the local director fluctuations along and transverse to the pitch axis to the microscopic properties of the constituents. We show that this correlation length diverges at the compensation point where the global twist vanishes.

From an experimental point of view, it would be highly desirable to dispose of model systems in which the molecular chirality (e.g. the microscopic pitch) can be carefully controlled. These would facilitate a systematic investigation of the relation between the micro- and mesoscale chirality and identify the presence of compensation points, cholesteric sense inversions and non-monotonic trends in the pitch versus particle concentration. Interesting opportunities lie in the application of filamentous phages to generate rod-shaped particles with tunable persistence length and chirality Barry et al. (2009); Dogic (2016), or in the self-assembly of chiral fibres of stacked organic compounds Engelkamp et al. (1999) or inorganic nanoparticles with bespoke shape and interactions Gao and Tang (2011).

Acknowledgements

The authors acknowledge helpful discussions with Paul van der Schoot. This work was funded by a Young Researchers (JCJC) grant from the French National Research Agency (ANR).

Appendix: Asymptotic estimates for the elastic moduli

Here we present asymptotic estimates for the Frank elastic moduli, K1 (splay), K2 (twist), K3 (bend) and K∗, that feature in the director fluctuation spectra. The corresponding microscopic expressions are very similar to Eq. (10). Fixing the reference director orientation along the x-axis of the laboratory frame (see Fig. 1(a)) we formulate Allen et al. (1993)

βK1

=ρ202⟨⟨w1zw2zM2˙f0(^ω1)˙f0(^ω2)⟩⟩,

βK3

=ρ202⟨⟨w1zw2zM(x)2˙f0(^ω1)˙f0(^ω2)⟩⟩,

βK∗

=ρ202⟨⟨w1xw2xM2˙f0(^ω1)˙f0(^ω2)⟩⟩,

(31)

where

M(x)2

=−∫dΔr(Δr⋅^x)2Φ(Δr,^ω1,^ω2)

=16L4D[(^ω1⋅^x)2+(^ω2⋅^x)2].

(32)

The elastic constants depend primarily on the achiral hard core of the particles and are assumed unaffected by the weak chirality imparted by the chiral potential Eq. (12). For strongly elongated hard rods Onsager’s theory can be invoked. Approximate analytical results can be obtained by employing a simple Gaussian test function for the ODF applicable to the regime where the local degree of nematic order is asymptotically large. The details of the analysis are outlined in Odijk’s paper Odijk (1986) and the asymptotic expressions for the elastic moduli are as follows